Family 4: moment maps, Monge–Ampère, and Stein kernels

KLS is now proved, by three different arguments compared in Chapter The proofs of KLS compared; all three start from the quadratic estimate of this family. This chapter describes what the moment map controls on its own and what it does not give without them.

Object followed. The Hessian $H=\Hess\varphi$ of the moment-map potential of $\mu$, in the coordinates in which $\mu$ is the pushforward of its own moment measure.

What it buys. The July 2026 moment-map estimates isolate a useful static component of the problem. In moment-map coordinates the statement “$\mu$ is isotropic” becomes “$\E_\nu H=I$”, and the Monge–Ampère equation, differentiated twice, produces two manifestly positive semidefinite source terms. Exploiting that positivity yields a sharp bound on $\E\Tr(BHBH)$ for every constant symmetric matrix $B$ — from which the sharp thin-shell constant, the sharp third-moment tensor, and the all-quadratic Poincaré inequality all follow.

This chapter presents the mechanism. The two consequences used elsewhere in this manuscript are Theorem 25.1 (Chapter The static quadratic-chaos input and the two-tail obstruction) and Proposition 26.1 (Chapter Small-time operator-norm control of the covariance); they are not restated here.

Moment-map coordinates¶

For a centered, full-dimensional $\mu=e^{-V}\dd x$, the moment-measure theorem supplies an essentially unique convex $\varphi$ with

Cordero-Erausquin & Klartag, 2015Klartag, 2014. Set

Change of variables in (4.1) is the Monge–Ampère equation

If $Y\sim\nu$ and $X=\nabla\varphi(Y)$, then $X\sim\mu$, and integration by parts under $\nu$ gives

when $\mu$ is isotropic. So “average Hessian equals identity” replaces isotropy in moment-map coordinates. This is the structural payoff: isotropy, which Chapter Family 1: classical needle localization showed is not inherited by needles, becomes a single linear identity for the object being estimated.

The moment-potential regularity and global gradient-image statement are Theorem 1.1 of Berman & Berndtsson, 2013; the target-coordinate Stein identity and weak zero-flux formulation are Theorem 2.3 of Fathi, 2019. This compact-target class is the published regularity input used by the approximation closure in Proposition 15.1.

Differentiating Monge–Ampère¶

Introduce the elliptic operator associated with the Hessian metric,

which is symmetric in $L^2(\nu)$:

Differentiating (4.3) twice gives the key identity

Both terms on the right are positive semidefinite: the first by convexity of $V$, the second because it is a Gram matrix of normalized third derivatives. This positivity is the PDE source of the new estimate; everything below is a way of harvesting it.

The fixed-matrix Hessian bound¶

Let $B\succeq0$ and define

Applying $\calL$ to $S_B$, integrating against $\nu$, and using the two positive sources of (4.9) yields

The nontrivial point is a third-derivative comparison. At a fixed point, normalize $H=I$ and diagonalize $B=\diag(b_i)$; writing $T_{ijk}=\varphi_{ijk}$, the difference between the two sides reduces to

Next apply Brascamp–Lieb entrywise to $F=B^{1/2}HB^{1/2}$. Since $\E F=B$ by (4.4),

and the Brascamp–Lieb energy of $F$ is exactly $D_B$, so

the last step by (4.11). Rearranging gives the main new theorem inside Letwin’s proof.

For indefinite $B$ the statement still holds: diagonalizing gives the pointwise bound $\Tr(BHBH)\le\Tr(\abs BH\abs BH)$, and (4.15) applies to $\abs B\succeq0$.

The $B=I$ case of (4.15) is the moment-Hessian estimate shared with the contemporaneous Chen–Klartag preprint.

The Stein kernel and the $H^{-1}$ inequality¶

Fathi observed that the moment-map Hessian, read in target coordinates,

is a positive symmetric Stein kernel for $\mu$ Fathi, 2019:

Combining (4.20) with the $H^{-1}$ calculus of Section The $H^{-1}$ norm and the Barthe–Klartag inequality gives, for a linear function $\ell_v(x)=\inner xv$,

By Remark 3.1, quadratic test functions have linear derivatives, so (4.21) can be fed into the Barthe–Klartag inequality (3.5). This is the junction at which the two families meet.

The noncommutativity trick¶

There is one genuine obstacle between (4.15) and a quadratic Poincaré inequality. For $q_M(x)=\inner{Mx}x$, applying (4.21) directly produces

which is not the expression controlled by (4.15), because $M$ and $\tau_\mu(X)$ need not commute.

First work on a regular target and suppose $M$ is invertible. The resolution is a change of variables. Write $M=\operatorname{sgn}(M)\abs M$ and set $Z=\abs M^{1/2}X$, with $\eta$ the law of $Z$. Then

and the Stein kernel transforms by congruence,

so that

by Theorem 4.2 applied with $B=\abs M$. The conjugation has moved $M$ inside the trace in exactly the pattern (4.15) controls.

The law $\eta$ is centered and log-concave, but generally not isotropic; the Barthe–Klartag $H^{-1}$ inequality requires no isotropy. Apply it to $F(z)=\inner{\operatorname{sgn}(M)z}z-\Tr M$, whose derivatives are centered. Since $M$ is invertible, $\operatorname{sgn}(M)$ is orthogonal, giving

For singular $M$, let $P_0$ project onto its kernel and apply this bound to $M_\varepsilon=M+\varepsilon P_0$, $\varepsilon>0$. Then

Variance therefore passes to the limit. Finally, Gaussian smoothing, convex truncation and affine normalization approximate any isotropic log-concave law by regular targets with convergence of moments through degree four. This transfers the quadratic estimate without requiring convergence of moment Hessians. Isotropy gives $\E\abs{\nabla\inner{MX}X}^2=4\Tr(M^2)$, yielding the formulation in Theorem 25.1, with constant 2 attained by products of centered exponentials at $M=I$.

The reduction from there to $\kappa_n\le2\sqrt2$ is short and purely algebraic; it is carried out as Proposition 26.1. The further bridge (0.14) gives the general bound Theorem 4.6, Theorem 1.1 of the preprint.

The constant $2\sqrt2$ is not sharp, and the moment-map approach says exactly what would sharpen it. On the regular moment-map class, Corollary 16.2 turns any bound $\E[\tau^2]\preceq c\,\Id$ into the directional third-moment bound $\norm{T_3(a)}_{\HS}\le2\sqrt{c-1}$, so the sharp linear test of the moment-Hessian inequality, $c=2$ (Conjecture 16.2), would give the sharp $\kappa_n\le2$, attained by products of centered exponentials. The implication runs one way only: a sharp third-moment bound does not return that sharp linear test, because the high-mode remainder of Lemma 16.2 is not zero off the cone axis.

The bridge must respect a restriction on the observation time. For a regular isotropic law, put $P=\CP(\mu)$. Gaussian observation, conditional variance, the Lipschitz-variance comparison and improved Lichnerowicz give

The factor $2+tP$ prevents using the whole time range of covariance control without checking the initial spectral gap. Set $T=a/(\kappa_n^2\log n)$ and $t=\min\{T,P^{-1}\}$. The fixed-time covariance estimate bounds the expectation by a universal constant, while $tP\le1$. If $t=T$, the result is $P\lesssim\kappa_n\sqrt{\log n}$; if $t=P^{-1}$, it is $P\lesssim\sqrt P$, hence a universal bound. Regular approximation and whitening pass the estimate to all isotropic log-concave laws. Combining Proposition 26.1 with Theorem 25.1, and then the two-sided Cheeger comparison, gives the two exponents above. This argument does not use Theorem 26.2 and retains the residual logarithmic loss.

What it does not reach alone. Control of $\E\inner{\tau_\mu(X)\nabla f(X)}{\nabla f(X)}$ for an arbitrary $f$, in place of $\E\Tr(BHBH)$ for a constant matrix $B$. Brascamp–Lieb in moment-map coordinates already gives

so the bound $\E\inner{\tau_\mu\nabla f}{\nabla f}\lesssim\E\abs{\nabla f}^2$ would give KLS within this family. The identity $\E\tau_\mu=I$ of (4.4) is not sufficient for this, because $\tau_\mu(X)$ can correlate with $\nabla f(X)$. Letwin controls a deterministic $B$; the missing theorem must handle an $X$-dependent matrix or direction field.

Where this family meets the alternative mechanisms. It supplies two of them. The moment map (Chapter The moment map: the deterministic inequality) attacks the gap above head-on, replacing constant multipliers by test-dependent Haar fields, and its endpoint — the target inequality $\mathrm{CMH}(4)$ for the constant $\CMH$, which everything in that mechanism serves to prove — is defined in these coordinates. The fixed eigenfunction uses the family’s quadratic control as the estimate it feeds its whitened posterior tensor to. The fixed-matrix bound (Theorem 4.2) is therefore the single literature input both mechanisms are trying to make adaptive.

Sources¶

The imports follow the first versions of the preprints of Letwin Letwin, 2026 (arXiv:2607.24164v1), whose Theorems 2.5, 1.2 and 1.1 are Theorem 4.2, Theorem 25.1 and Theorem 4.6, and of Chen–Klartag Chen & Klartag, 2026 (arXiv:2607.23307v1), Theorems 1.5, 1.1 and 1.2 with Corollary 1.3 for convex bodies. In both, the passage from regular targets to general log-concave laws goes through convergence of polynomial moments under Gaussian smoothing, convex truncation and affine normalization; it asserts no convergence of the moment-map Hessian. The full third-tensor norm bounded by Chen–Klartag is distinct from the directional parameter of Proposition 26.1.